Evidence doesn't have to establish the truth for which it is used to establish in order for what is used as evidence to be evidence. With proof, however, it's different.
You seem to say that evidence not need increase the probability of the conclusion to 1 ("establishing the truth") but proof does. Maybe. But I disthink so. I use "proof" more broadly than that and most people do like me. A proof is merely very strong evidence.
The limited notion of "proof" that you seem to be talking about is the proof of mathematics and logic.
Isn't that so?
If I provide my identification as evidence that I am who I say I am, then not only have I provided evidence that I am who I say I am, but I have also provided proof that I am who I say I am. But, this is a simple case.
I don't know what identification you provided, so I will not say that you did provide proof. However in the limited sense of "proof", you did not provide proof since the evidence did not increase the probability of the conclusion to 1. No identification that I know of could ever be proof in the limited sense.
If my brother provides my identification as evidence that he is who he says he is, then yes, he has provided evidence, and yes, he has provided evidence that he is who he says he is (believe it or not!), but although he has provided the evidence as proof that he is who he says he is, he didn't actually provide proof that he is who he said he is because my identification doesn't prove that he is me.
So, in the simple case, evidence serves as proof, but in the complex case, the evidence does not serve as proof. In both cases, evidence is provided, but only in the simple case was there actual proof.
Depending on the identification used, it may or may not provide proof for your brother. But it did not provide proof in the limited sense.
What's interesting to me is that although we can have evidence for what doesn't exist, we can't have proof for what doesn't exist.
So you say. Is this the "You can't prove a negative."? I might disagree depending on what is meant by that.
To recap. There are (at least) two sense of "proof".
- The general sense where "proof" means (very) strong evidence. In symbols(ish). Pr(p|proof)>>Pr(p|evidence). We may refer to this as g-proof for general proof.
- The limited sense used in mathematics and logic where proof is what "establishes the truth of the conclusion". Formally Pr(p|proof)=1. We may refer to this as l-proof for limited proof.
Alternatively we could stipulate in this context only to use proof in the limited sense (l-proof). Another alternative is that we could refer to them as weak and strong proof or with adverbs weakly proves and strongly proves.
---------- Post added 11-16-2009 at 07:14 PM ----------
One cannot believe a contradiction, as far as I know.
You may want to learn about paraconsistent logic (SEP